Figure 2

Pathogen invasion conditions with time-invariant dynamics (a,b) or seasonal variations of the transmission parameters \(\eta (t)\) and \(\beta (t)\) (c,d). (a,b) The DFE and the EE collide and exchange their stability in correspondence of the black thick line (i.e. \(R_0=1\)). Dashed curves represent the contour levels of the prevalence at the EE, \({\bar{I}}_e/H\), evaluated via numerical simulations of the model 1–5, with \(I(0)=10\) and \(S(0)=H-I(0)\). (a) Prevalence curves for varying exposure rate \(\beta _0\) vs \(R_0\). (b) Prevalence curves for varying probability to thawing-released spores \(\eta _0\) vs \(R_0\). (c) Endemicity thresholds are represented in the \({\tilde{R}}_0\)–\(\epsilon _{\eta }\) parameter space for different values of seasonality \(\epsilon _{\beta }\). The corresponding \({\overline{R}}_0\) is also displayed (coloured markers). (d) Endemicity curves in case of lagged signals of \(\beta (t)\) and \(\eta (t)\) for different values of \({\tilde{R}}_0\). Parameter values as in Table 2. Other parameters: \(\beta _0=1\) (b,d), \(\eta _0=0.5\) (a). In panels (a,c) we set a range of variability for \(R_0\) and \({\tilde{R}}_0\) (between 0 and 100) and consequently calculated \(\theta\).